The car of mass
represented
in the diagram on the left is moving at the velocity
in a curve of
radius
. If the car
is not skidding, the friction coefficient between the road and the tires
is
.
As always, in order for the car to move in the curve, one of the
forces acting on the car must be the one responsible for the centripetal
force. On the car, two forces act on the vertical direction, the weight
of the car
and the reaction of the surface to this force or normal force
.
Clearly, these two forces cancel each other. However, there is a
horizontal force directed to the center of the circle, this is the
static frictional force between the road and the tires of the car
.
This force appears because the tendency of the car (tires) is to
maintain the same state of motion (law
of inertia) which corresponds to the car moving in a straight line.

Therefore, Newton's second law of motion applied to the car
is
with .
On the other side of the equation, the acceleration is the centripetal
acceleration,
with
the direction of the acceleration toward the center of the circle. Combining
these relations, the following equation is obtained
. In
this equation, the necessary static frictional force increases as the square of
the velocity. Remember that the magnitude of the static frictional force adjusts
to the necessary value to prevent the object from skidding but only to a maximum
limit. In the limit,
where
.
Thus, the maximum value of the frictional force can be substituted in the
previous relation to become
from
where .
The last equation can be used to solve problems such as, for a given coefficient
of static friction, what is the maximum velocity at which a curve may be taken,
, etc.